I assume you are talking about the PROOT command, or the Solve Poly interface? This is what is found in the source there:

** The algorithm is Laguerre iteration with stepsize control
** and is based upon the ZERPOL program contained in the thesis
** entitled "A Zero Finding Algorithm Using Laguerre's Method"
** by Brian T. Smith (directed by W. Kahan).

** The implementation is a modified version of the HP-71 Math
** ROM PROOT runtime execution code implemented by Laurence Grodd.

I was looking at the HP-71B Math Pac manual and it does mention the Laguerre method as being used for that ROM (and I am sure for many other HP calculators). I have a Matlab function that implements of the Laguerre method and it works very well ... so the method is good, even though my implementation is bare bone -- not scaling or other programming tricks.

Regarding PROOT on HP calculators, the following post from Bill Wickes on comp.sys.hp48 back in 1992 shows that it was quite easy to port HP 71B assembler functions to the HP-48:
HP 48 Polynomial Rootfinder

Given this, I'm a bit surprised that the results for the second test on the HP 71B (+ Math Pac) and the HP 48GX (same CPU, same algorithm, same assembler code?) or the 49G+ in the following post on this forum are somewhat different (The Turtle (HP-71B) and the Hare (HP49G+) [LONG]):

0.999999999944, 1.312E-12 for the HP 71B
0.999999994032, 2.066E-12 for the 48GX
0.999999994031, 1.441E-12 for the 49G+

... the HP 71B while being the oldest machine is the closest to the exact result (1,0)!